Sudoku Strategies and Solving Techniques
Scanning the board for obvious placements will carry you through easy and most medium puzzles. Beyond that you need techniques that reason about candidates — the digits still possible in a given cell — rather than about finished numbers. The techniques below are ordered roughly by how often they're useful, so learning them in this order gives you the most benefit soonest.
Candidates: The Foundation
A cell's candidates are every digit from 1 to 9 that hasn't already been used in that cell's row, column or box. A cell with a single remaining candidate is solved. Every technique on this page is ultimately a method for removing candidates until only one is left somewhere.
Experienced solvers rarely write out full candidate lists for all 81 cells — it's slow and it clutters the board. Instead they track candidates for one digit or one region at a time, which is what the techniques below are built around.
Naked Single
The simplest technique of all: a cell with exactly one remaining candidate. If a cell's row, column and box between them already contain eight distinct digits, the ninth is forced into that cell.
Naked singles are usually the last step in a chain rather than the first — each number you place tends to create a few more of them nearby, which is why solving a sudoku often accelerates dramatically towards the end.
Hidden Single
A hidden single is a digit that has only one possible home within a row, column or box — even though the cell it lands in may still have several candidates of its own. It's "hidden" because you find it by looking at the group as a whole rather than at the cell.
Example: in one 3x3 box, five of the empty cells could hold a 7 according to their own candidates — but four of them sit in rows that already contain a 7 elsewhere. Only one cell survives, so it must be the 7. Hidden singles are the workhorse technique of medium and hard puzzles, and most puzzles below expert level can be solved with hidden and naked singles alone.
Naked Pairs, Triples and Quads
If two cells in the same group have exactly the same two candidates — say both can only be 4 or 9 — then between them those two cells will consume both the 4 and the 9. It doesn't matter which order. That means 4 and 9 can be eliminated from every other cell in that group.
The same logic scales up. Three cells sharing three candidates form a naked triple; four cells sharing four candidates form a naked quad. The cells don't each need every candidate — three cells holding {2,5}, {5,7} and {2,7} still lock up 2, 5 and 7 between them.
Hidden Pairs
The mirror image of a naked pair. If two digits can each only go in the same two cells within a group, those two cells must hold those two digits — so every other candidate can be removed from both cells, even if they were each carrying five or six candidates before.
Hidden pairs are easy to walk straight past, because nothing about the cells looks unusual until you check where each digit can go.
Pointing Pairs and Box-Line Reduction
These two techniques both exploit the overlap between a box and a line, and they're the first real jump in difficulty.
Pointing pairs
If, within one 3x3 box, every possible position for a digit lies in the same row, then that digit must end up in that row somewhere inside the box. It therefore cannot appear in that same row anywhere outside the box — so you can eliminate it from the six cells in the rest of the row.
Box-line reduction
The reverse direction. If every possible position for a digit within a row falls inside a single box, that digit must be in the box on that row — so it can be eliminated from the other six cells of the box.
Both are simple to check once you're in the habit, and they unlock a great many puzzles that otherwise look stuck.
X-Wing
The first genuinely advanced pattern. Look for a digit that has exactly two possible positions in each of two different rows, and where those positions fall in the same two columns. Those four cells form the corners of a rectangle.
Whichever way the digit resolves, it must occupy one corner in each row, and therefore one cell in each of the two columns. That means the digit can be eliminated from every other cell in both of those columns. The same works with rows and columns swapped.
X-Wings appear rarely, but when a hard puzzle refuses to yield to anything else, one is often hiding somewhere.
Practical Advice
- Work one digit at a time. Sweeping the whole board for a single digit is far more reliable than scanning cell by cell, and it's how hidden singles reveal themselves.
- Re-scan after every placement. Each number you write changes the candidates of up to twenty other cells. Solvers get stuck mostly because they're working from a mental picture that's several moves out of date.
- Never guess. If a puzzle has one solution, it can be solved by logic. Being stuck means you haven't found the right group to look at yet — not that the puzzle needs a coin flip.
- Change your view when stuck. Switch from scanning rows to scanning boxes, or from looking at cells to looking at digits. Most stalls break the moment you change perspective.
- Save your hint for a real wall. You get one per puzzle here, and it awards no points — it's there to keep a run alive, not to solve the board for you.